This no longer works, as of September 2015 (and now Oct.) No matter what I do, the Mac version of Chromium will not do Flash. Not sure what version this broke in (I have been bouncing around between various Dev and Canary builds). You can even manually edit ~/Library/Application Support/Chromium/Default/Preferences to point directly at the PPAPI Flash plugin, after copying it out of Chrome and putting it into /Applications/Chromium.app/Contents/Versions/[version number]/Chromium Framework.framework/Internet Plug-Ins, mirroring the Chrome installation and Preferences file down to the last details in the plugins section of the JSON in the Preferences file, and it will not work. Also tried pointing it explicitly at the copy inside Chrome, and at the newer version installed by Adobe in "/Library/Internet plugins", to no avail. Guess I'm just going to relegate Chromium to a dev browser that I don't use day-to-day. Too much hassle to keep having to flip over to another browser and reload the same URL on every other site I go to because their news videos, Flash-driven menus, etc., don't work in Chromium any longer. This is definitely NOT an Adobe installer issue, it's a Chromium issue. I've even tried tricks that work with the Linux versions:
~/.bash_profile:
export PEPPER_FLASH_VERSION=$(grep '"version":' /Library/Internet\ Plug-Ins/PepperFlashPlayer/manifest.json | /opt/local/bin/grep -Po '(?<=version": ")(?:\d|\.)*')
/etc/chromium-browser/default and /etc/chromium/default:
CHROMIUM_FLAGS="--ppapi-flash-path=/Library/Internet\ Plug-Ins/PepperFlashPlayer/PepperFlashPlayer.plugin --ppapi-flash-version=$PEPPER_FLASH_VERSION"
No effect at all. I've tried a clean install of Flash, with Chromium set as the default browser, to see if Adobe's own installer would do it right. Didn't work. If you go to chome://flash it says it is not installed, every time. Weirdly, I find that the cache directories in "~/Library/Application Support/Chromium/Default/Pepper Data/Shockwave Flash" have material in them as recent as just a couple of days ago, despite the fact that I've not gotten the plugin to work in Chromium since early Sept.
If you have a version of Chromium that is presently working with Flash, you should probably archive a backup of the app, its "~/Library/Application Support/Chromium" directory, and the ~/Library/Preferences/org.chromium.Chromium.plist file. My next experiment will be to revert back to a Stable build from around June or so, and see if I can get that to work, then try upgrading to a Dev build without breaking the Flash support.